Is it possible to export a 3D object made in Illustrator to Blender as a mesh?
I'm guessing that what you see is something like this.



I don't know what Illustrator is capable of, because I do not have it, and I have never used it. However, Blender can clean it up, by use of the Remesh modifier.

It has three modes, Sharp, Smooth, and Blocks. They work a bit differently, and which one you use, depends on what you want the resulting mesh to look like.



The Octree Depth setting sets the resolution. The Scale setting twaeks the resolution on a more fine-grained level. Smooth Shading should be self explanatory. Remove Disconnected Pieces removes small pieces of geometry that are not connected to the main mesh, and the Threshold value controls how small/big pieces get removed.
The Sharp mode also has a Sharpness setting; lowering it makes the mesh smoother and increasing it makes it sharper.

N.B. If your SVG has several unconnected parts, do not join them into a single object until after remeshing, or you will lose individual materials. Moreover, vertex groups and UV maps are not retained by the remesh.